Airport Gate Scheduling with Time Windows

Transcription

1 Artificial Intelligence Review (2005) 24:5 31 Springer 2005 DOI /s Airport Gate Scheduling with Time Windows A. LIM 1, B. RODRIGUES 2, &Y.ZHU 1 1 Department of IEEM, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong; 2 School of Business, Singapore Management University, 469 Bukit Timah Road, Singapore (*Author for correspondence, br@smu.edu.sg) Abstract. In contrast to the existing airport gate assignment studies where flight have fixed schedules, we consider the more realistic situation where flight arrival and departure times can change. Although we minimize walking distances (or travel time) in our objective function, the model is easily adapted for other material handling costs including baggage and cargo costs. Our objectives are achieved through gate assignments, where time slots alloted to aircraft at gates deviate from scheduled slots minimally. Further, the model can be applied to cross-docking optimization in areas other than airports, such as freight terminals where material arrival times (via trucks, ships) can fluctuate. The solution approach uses insert and interval exchange moves together with a time shift algorithm. We then use these neighborhood moves in Tabu Search and Memetic Algorithms. Computational results are provided and verify that our heuristics work well in small cases and much better in large cases when compared with CPLEX solver. Keywords: aircraft gate scheduling, tabu search, memetic algorithm 1. Introduction Gate scheduling for flights is a key activity at airports. Aircrafts are assigned to terminal, called gates, or ramp positions for the duration of a time period during which passengers and aircrafts are processed. Gate availability and times of arrivals/departures (as given by an estimated time of arrival/departure or ETA/ETD for each flight) can change during the course of the planning horizon due to operational contingencies (for example, delays, air traffic control). Such changes in turn impact on gate availability, since gates may become unavailable when required. A familiar scene at airports these days is when arriving flights are forced to wait on the ramp, sometimes for a long time, before finally proceeding to their gate because the gate is occupied by another aircraft. Airlines themselves have preferences for gates (within terminals) and time slots for commercial and operational

2 6 A. LIM ET AL. reasons. For example, gates close to each other are preferred by airlines for transfer passengers while adherence to published time slots allows for smooth aircraft processing and is good for airline image. Both gates and time slots are scarce and expensive resources. Airport operators plan for better airport design and increasing gates to cope with increasing demand. For the airline, changes in time slots (including those resulting from poor gate assignment) incur additional operations costs (for example, landing and parking fees, staffing) and inconveniences for its passengers. Gate assignment affects the quality of service an airline provides its passengers. Good gate assignment can help airlines keep to published schedules by reducing gate delays. Good gate assignment can minimize distances (times) passengers are required to walk from gate to gate or from gates to exits/entrances. In the case of connecting passengers, these distances are crucial since such passengers can easily miss flights, especially for connections with small minimum connecting times. Unavailability of gates leads to arrival/departure time changes that have a roll-on effect on other ground services. These include fuelling, aircraft cleaning, cargo loading, catering uplift, baggage handling and passenger handling, some of which require setup times. The impact is exacerbated when airlines cut down on turnaround/transit times for delayed flights to catch up with their schedules. While airports are faced with growing civil air traffic, the complexity of gate assignments has increased significantly in the past decades. Because of the large number of flights handled and the dynamic nature of the problem, scheduling has become more difficult. This has made it more important for airport operators to use gates in the best possible way. Gate assignments are planned for seasonal flight schedules and, given such a schedule, daily plans are prepared and updated on the actual day of operation where they are frequently changed to accommodate delays and disruptions in the planned schedule in what can be called reactive scheduling which is frequently sub-optimal. The purpose of this work is to address this problem and examine scheduling flights where, in contrast with previous work on the subject where schedules are fixed, we allow for flights to arrive and depart within time windows. This model reflects the dynamic nature of gate assignments and is useful in applications. The model is new. It extends the classical gate assignment problem, where only walking distances are considered and schedules are fixed. In addition, the model can be

3 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 7 used to handle operational costs in material handling like baggage and cargo handling. This work is organized as follows: in the next section, we discuss previous work on gate assignment. In Section 3, we give a model of the problem. Tabu search (TS) and memetic algorithms (MA) are developed in Section 4. We provide computational results in Section 5. In Section 6, we discuss its application to the cross-docking problem. The conclusion is given in Section Previous Work One of the first attempts to use quantitative means to minimize intra-terminal travel into a design process was given by Braaksma and Shortreed (1971). The assignment of aircraft to gates, which minimizes travel distances, is an easily motivated and understood problem but a difficult one to solve. The total passenger walking distance is based on passenger embarkation and disembarkation volumes, transfer passenger volumes, gate-to-gate distances, check-in-togate distances and aircraft-to-gate assignments. In the gate assignment problem, the cost associated with the placing of an aircraft at a gate depends on the distances from key facilities as well as the relations between these facilities. The basic gate assignment problem is a quadratic assignment problem and shown to be NP-hard in Obata (1979). Babic et al. (1984) formulated the gate assignment problem as a linear 0-1 Integer Programming (IP). A branch-and-bound algorithm is used to find the optimal solution, where transfer passengers are not considered. Mangoubi and Mathaisel (1985) used an LP relaxation of an IP formulation and greedy heuristics solve the problem of Babic et al. (1984) with the difference that their model includes transfer passengers. Bihr (1990) used 0-1 Linear Programming (LP) to solve the minimum walking distance gate assignment problem for fixed arrivals in a hub operation where a simplifying formulation as an assignment problem is employed. Wirasinghe and Bandara (1990) considered, additionally, the cost of delays and used an approximation procedure in their analysis. Yan and Chang (1998) proposed a network model that is formulated as a multi-commodity network flow problem. An algorithm based on the Lagrangian relaxation was developed to solve the problem. In a different approach, simulation analysis was used by Baron (1969) to analyze the effects of passengers walking distance resulting from different gate-use strategies where both local and

4 8 A. LIM ET AL. transfer passengers are considered. Other work based on simulation models can be found in, for example, Cheng (1998a, b). Since the gate assignment problem is NP-hard, various heuristic approaches have been used by researchers. Haghani and Chen (1998), proposed a heuristic that assigns successive flights to be parked at the same gate when there is no overlapping. In the case where there is overlapping, flights are assigned based on shortest walking distances coefficients. More recently, Xu and Bailey (2001) provide a TS meta-heuristic to solve the problem. The algorithm exploits the specific properties of different types of neighborhood moves, and creates highly effective candidate list strategies. Also, recently, work has focused on the over-constrained airport gate assignment (Ding et al. 2003a, b), where there is an excess of flights over gates. The objective there was to minimize the number of flights without any gate assigned (i.e. those left on the ramp) and the total walking distance. A greedy algorithm was proposed to minimize the number of flights without any gate assigned and an interval exchange neighborhood search was designed and applied within metaheuristic frameworks, such as TS, Simulated Annealing, and their hybridization. 3. Optimization Model In the previous work on gate assignment, flight schedules have been fixed. This means that, once a flight lands, it is assigned to a gate immediately. In reality, however, no gate may be available. We assume that the duration of transit of each flight in the airport is fixed, then we can allow this transit duration to slide in a time window. For example, suppose a flight requires a transit duration of two hours and is assigned a time window of h, then the flight can occupy gates between 0300 and 0500 h, or 0330 and 0530 h, or 0400 and 0600h, and so on. Any two flights can be assigned to the same gate if their actual gate occupation times do not overlap. Consider the three flights: Flight 1: Time window is h, transit duration is 3 h; Flight 2: Time window is h, transit duration is 4 h; Flight 3: Time window is h, transit duration is 5 h. If we need to assign these to one gate, Figure 1 shows three possible assignments all of which are feasible.

5 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 9 Time Window of Flight 2 Time Window of Flight 1 Time Window of Flight 3 Flight 1 Flight 2 Flight Assignment 1 Time Window of Flight 2 Time Window of Flight 1 Time Window of Flight 3 Flight 1 Flight 2 Flight Assignment 2 Time Window of Flight 2 Time Window of Flight 1 Time Window of Flight 3 Flight 1 Flight 2 Flight Assignment 3 Figure 1. Three feasible assignments. Since it is preferred that flights be assigned a gate as soon as possible, we consider penalties proportional to the delay time, where we allow different flights to have different unit penalties. Reasons for the latter could be that the penalty for a large aircraft with many passengers could be higher that for a small aircraft with a few passengers or because ground handling costs vary from airline to airline. Flights with short connecting times could also attract higher penalties. For example, if a i is the start point of flight i s time window, c i is the actual arrival time at the gate, and p i is the unit penalty, then the delay penalty is deemed to be p i (c i a i ). Our objective is to minimize the sum of the delay penalties and the total walking distance (travel time). The problem is NP-hard since it is a generalization of the classical airport gate assignment problem.

6 10 A. LIM ET AL The gate assignment model with time windows The airport gate assignment problem with time windows can be represented as follows. To simplify the problem, we treat all time values as the discrete points in a time horizon. Parameters n: number of flights m: number of gates a i : start point of flight i s time window b i : end point of flight i s time window d i : gate occupation duration of flight i p i : unit delay penalty of flight i w kl : walking distance between gate k and gate l f ij : number of passengers between flight i and flight j M: a sufficiently large number Decision variables x ik {0, 1}: 1 iff flight i is assigned to gate k; 0 otherwise c i N: the time when flight i starts to occupy a gate; y ij {0, 1}: 1 iff flight i departs no later than flight j lands; 0 otherwise z ij kl {0, 1}: 1 iff flight i is assigned to gate k and flight j is assigned to gate l Minimize subject to n n m i=1 j=1 k=1 l=1 m f ij w kl z ij kl + n p i (c i a i ) i=1 m x ik = 1, 1 i n (1) k=1 z ij kl x ik, 1 i, j n, 1 k,l m (2) z ij kl x jl, 1 i, j n, 1 k,l m (3) x ik + x jl 1 z ij kl, 1 i, j n, 1 k,l m (4) c i a i, 1 i n (5)

7 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 11 c i b i d i, 1 i n (6) (c i + d i ) c j + y ij M>0, 1 i, j n (7) (c i + d i ) c j (1 y ij )M 0, 1 i, j n (8) y ij + y ji z ij kk, 1 i, j n, i j, 1 k m (9) We note that c i is optimally the ETA (Estimated Time of Arrival) of Flight i. The values assigned to time windows [a i,b i ] can be determined on statistical data collected for particular flights. The closer these are to [ETA, ETD] (ETD stands for Estimated Time of Departure), the better since this reflects flights close to schedule and hence more likely to take up pre-assigned preferred gates. Constraints (1) ensures that each flight must be assigned to exactly one gate. Constraints (2) (4) jointly define the variable z (a similar definition can be found in Xu and Bailey 2001). Constraint (5) and (6) ensure that the flight must land and depart within the specified time window. Constraints (7) and (8) define the properties of the variable y: constraint (7) indicates that y ij = 1if(c i + d i ) c j, which means y ij = 1 when flight i departs before or right at the time when some gate opens for flight j; constraint (8) indicates that y ij = 0 if (c i + d i )>c j, which means y ij = 0 when flight i departs after some gate opens for flight j. Finally, constraint (9) specifies that one gate cannot be occupied by two different flights simultaneously. Since in this IP model, the objective function and constraints are linear, it is not difficult to use standard IP solvers, such as CPLEX, to find optimal solutions for some instances. However, the number of inequalities is large even for medium size instances, such as constraint (4), which has O(n 2 m 2 ) number of inequalities. Hence the running time of branch-and-bound algorithm of the IP solver would be long, as is verified in the experimental results section Other models In the above model, we considered minimizing passenger walking distance (travel time) allowing time windows for aircraft. As a proxy measure of good passenger handling, this is only one way to measure cost. In addition to passenger handling costs, operational costs in airport terminals result from other material handling activity, which include baggage and cargo handling. Airport terminals are similar to freight terminals where material is transferred from trucks door to

8 12 A. LIM ET AL. door using different material handling systems. In airports, freight can include passengers, baggage and cargo. Further, passengers can be differentiated into transit and embarking/disembarking passengers, or by airlines, or by class of travel. Each of these groups require specialized handling where different costs can be applied. For example, transit passengers are generally more difficult to handle and require more attention from ground staff than do embarking/disembarking passengers. Baggage handling follows passenger handling closely since baggage follow passengers. Here, connecting baggage, especially baggage between different airlines, require special attention. Baggage can be loaded into containers or freely into holds, requiring different material handling systems (baggage trolleys, container loaders, forklifts, sorters). Cargo is loaded into containers (AVE s), pallets, igloos or freely into holds. Unloading and loading cargo require a number of different machines and specially trained staff. Cargo transfer between aircraft within short periods can be especially difficult and expensive since more equipments and staffs need to be deployed. When compared to passenger handling, baggage and cargo handling require longer setup times. The flow of passengers, baggage and cargo is dependent on terminal layout, geometry and handling systems. At the operational level, flow is greatly affected by actual gate assignments. Getting the preassigned gate can save a significant amount of time in terms of repositioning equipment and staff for baggage and cargo handling as well as other ground services. Short of this, getting gates that are close on the ramp to each other is next best for baggage and cargo transfer. The model developed above can be modified to address some of these costs. For example, for baggage handling, we can replace passengers by numbers of baggage to be transferred, and walking distances by ramp inter-aircraft distances to achieve best possible aircraft gate assignments. In the case of cargo transshipment, we illustrate with a simple example where the model can be adapted. Take the case where there are two categories of cargo, A and B, which require different material handling systems. With all parameters and decision variables as above, take fij A and fij B to be the cargo loads from flight i to flight j of type A and B respectively, c kl to be the cost of moving a unit of either type of cargo from gate k to gate l, and c A and c B be the unit cost of handling cargo of type A and B, respectively. A model to optimize gate assignments with cargo handling costs can then be written as:

9 i=1 j=1 k=1 l=1 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 13 Minimize n n m m [ f A ij (c kl + c A ) + fij B (c kl + c B ) ] z ij kl + n p i (c i a i ) i=1 subject to constraints (1) (9). This model, as an IP, is essentially the same as the gate assignment model with time windows given above. 4. Solution Approaches 4.1. The neighborhood search moves In Xu and Bailey (2001), three kinds of neighborhood moves are proposed which are used in the TS algorithm to solve the classical airport gate assignment problem. They are described as follows: Insert Move: Move a single flight to a gate other than the one it is currently assigned to. Exchange I Move: Exchange two flights and their gate assignments. Exchange II Move: Exchange two flight pairs in the current assignment. The two flights in the flight pair have to be consecutive. Those moves, however, are shown to be ineffective sometime (Ding et al. 2003a). In heuristics that are proposed in Ding et al. (2003a), an Interval Exchange Move is designed to replace the Exchange I Move and Exchange II Move. The Interval Exchange Move exchanges two flight intervals in the current assignment. A flight interval consists of one or more consecutive flights in one gate. The authors show that the TS which adopts Interval Exchange Move is superior to methods used in Xu and Bailey (2001). Here, we will use the Insert Move and Interval Exchange Move to devise our neighborhood search moves. We note that in the previous airport gate assignment model in Xu and Bailey (2001) and Ding et al. (2003a), gate occupation times of each flight are fixed; whereas they can occur in a time window here. We need to perform time duration shifting (here time duration refers to the gate occupied time for each flight) when performing the insertion or exchange. The

10 14 A. LIM ET AL. Time Window of Flight 4 Time Window of Flight 8 Time Window of Flight 2 Time Window of Flight 5 Gate 4 Flight 2 Flight 4 Flight 5 Flight ShiftLeft (4,2) Time Window of Flight 4 Time Window of Flight 8 Time Window of Flight 2 Time Window of Flight 5 Gate 4 Flight 2 Flight 4 Flight 5 Flight Figure 2. An example of ShiftLeft subroutine. details of our neighborhood search approach are introduced in the following sections Time duration shift subroutines In order to aid us to perform the Insert Move and Interval Exchange Move, we define the following auxiliary subroutines to perform the time duration shifting: ShiftLeft(k, i) shifts the time durations starting from the ith flight at gate k as left (earlier time) as possible. Each flight can be shifted to left until the start point of its time window or to a point it is constrained by the previous flight. Figure 2 illustrates an example. The subroutine ShiftLeft(4,2) shifts the time durations from second flight (Flight 4) at gate 4. Therefore, Flight 4 can be shifted to time 4 as 4 is the start point of Flight 4 s time window; Flight 5 can be shifted to time 8 as it cannot overlap Flight 4; Flight 8 can be shifted to time 12 as it is the start point of the time window. ShiftRight(k,i,t) shifts the time durations from the ith flight at gate k to right (later time) until the ith flight assigned to the gate at time t. Each flight has to be shifted to right to avoid overlapping. Figure 3 illustrates an example. The subroutine ShiftRight(4, 2, 6) shifts the time durations from second flight

11 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 15 Time Window of Flight 4 Time Window of Flight 8 Time Window of Flight 2 Time Window of Flight 5 Gate 4 Flight 2 Flight 4 Flight 5 Flight ShiftRight (4,2,6) Time Window of Flight 4 Time Window of Flight 8 Time Window of Flight 2 Time Window of Flight 5 Gate 4 Flight 2 Flight 4 Flight 5 Flight Figure 3. An example of ShiftRight subroutine. (Flight 4) at gate 4. Therefore, Flight 4 is shifted to time 6 as required; Flight 5 can be shifted to time 10 as it cannot be overlapped with Flight 4; Flight 8 remains unchanged. Note that we have to ensure the value t will not cause the shifted flights to violate time window constraints when the subroutine is used. AttemptShiftRight(k, i) returns the latest time that gate k can be occupied for the ith flight, if all the following flights are shifted as right as possible. Each flight can be shifted to right until the end point of its time window or to a point where it becomes constrained by the next flight. The algorithm attempts to perform the right shift from the last flight to the ith flight. Note this subroutine only returns a possible value instead of modifying the gate occupied time, which is different from the above-mentioned two subroutines. ShiftInterval(k,i,j,t) shifts the time durations from the ith flight to the jth flight at gate k until the ith flight assigned to the gate at time t. Each flight has to be shifted to avoid overlapping. It is almost the same as the subroutine ShiftRight(k,i,t) which is mentioned above. The difference is that this routine only shifts the durations between the ith and the jth flights, while the previous one considers all the durations after the ith flight.

12 16 A. LIM ET AL. AttemptShiftInterval(k,i,j,t) returns the end time of the j th flight if the time durations from the ith flight to the j th flight are shifted until the ith flight assigned to the gate at time t. It is similar to the subroutine ShiftInterval(k,i,j,t) except that it returns the end time instead of modifying c f to perform actual shifting. AttemptShiftIntervalRight(k,i,j) returns the latest time that gate k can be occupied for the ith flight, if the time durations from the ith flight to the jth flight are shifted as right as possible. The algorithm is identical to the subroutine AttemptShiftRight(k, i) except that it only shifts durations between the ith and the jth flights. Having introduced these subroutines, we can describe the algorithms for two neighborhood moves: Insert Move and Interval Exchange Move The Insert Move The Insert Move insert(i,k) (i, k ) moves a flight i, which is currently assigned to gate k, togatek (k k). In each move, i and k are randomly selected, where k is naturally obtained from the current solution. We need to know how to determine whether flight i can be inserted (assigned) to gate k and which is the appropriate place to insert it. In the classical airport gate assignment in Xu and Bailey (2001) and Ding et al. (2003a), feasibility is ensured by simply checking whether the current time duration of flight i overlaps with the flights assigned to gate k. In our problem, however, seeking a feasible neighborhood solution is much more complicated, because flight i can be shifted in its time windows and so can the flights in gate k. We first note that in our solutions all flights are assigned to gates as early as possible with respect to the current assignment. That is, each time duration cannot be shifted to left any more because it has reached the start point of its time window, or it is constrained by the preceding flight. This follows from minimizing the delay penalties. Suppose we want to insert flight i between flight u and flight v in the gate k. The first step is to check whether the gap between u and v can accommodate i. To make the gap as large as possible, we try to move flight u as far left as possible and flight v as far right as possible. But in fact, moving u to the left is not necessary according to the above-mentioned property in our solutions; moving v to the right causes cascading effects to move some flights that follow v. Thus, the

13 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 17 Figure 4. The Insert Move. subroutine AttemptShiftRight(k,v) will be invoked. Step 2 of Figure 4 illustrates this. The dashed arrow line means the time durations are only virtually moved to find the time point but not changed. We find the two end points of the gap, which are indicated t 1 and t 2 in Figure 4. The inequality t 1 + d i t 2 checks whether the gap can accommodate flight i. If insertion is possible, the actual move will be performed. First, the subroutine ShiftRight(k,v,t 1 + d i ) is invoked. We shift the flight v to the time point t 1 +d i because flight i will be placed at time point t 1 later. Step 3 of Figure 4 illustrates the process. The normal arrow line indicates that we actually perform the shift move and assignment.

14 18 A. LIM ET AL. Figure 5. Six time points in an interval. Finally, the time durations after flight i in gate k should be shifted left since flight i is removed and we always place the time durations as left as possible, as mentioned above. Hence the subroutine ShiftLeft(k, j), where j is the flight that follows flight i in gate k. Step 4 in Figure 4 illustrates this adjustment. Algorithm 1 describes briefly the Insert Move process. This move changes the value of objective function which is easily calculated. Algorithm 1 The Insert Move Algorithm succ false while not succ do generate i and k randomly, get the value of k for each flight v in gate k from left to right do if c v + d v >a v then t 1 the ending time of the flight preceding v t 2 AttemptShiftRight(k,v) if t 1 + d i t 2 then succ true ShiftRight(k,v,t 1 + d i ) remove flight i from gate k insert flight i into gate k at time t 1 ShiftLeft(k, j) {j is the flight that follows i in gate k} update the cost break end if end if end for end while The Interval Exchange Move The Interval Exchange Move (i 1,j 1,k 1 ) (i 2,j 2,k 2 ) exchanges the time durations from flights i 1 to j 1 in gate k 1 with flights i 2 to j 2 in gate k 2.

15 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 19 The idea was proposed in Ding et al. (2003a) to overcome a weakness of Exchange I and Exchange II moves in Xu and Bailey (2001). However, we need to modify the previous algorithm since the time durations can change here. In our implementation, i 1,j 1,i 2,j 2 are randomly generated, and then k 1 and k 2 can be obtained. Thus, similar to the above-mentioned Insert Move, we need to determine whether the two intervals are compatible for exchange. Consider an arbitrary interval [i 1,j 1 ]. There are six time points which are crucial for us to determine the compatibility. They are shown in Figure 5: t 11 : The end time point of the flight preceding i 1 ; t 12 : The latest start time point of the flight following j 1 ; t 13 : The start time point of flight i 1 ; t 14 : The end time point of flight j 1 ; t 15 : The latest start time point of i 1 if we shift interval [i 1,j 1 ]as right as possible; t 16 : The latest end time point of j 1 if we shift interval [i 1,j 1 ]as right as possible; These six time points can be calculated as follows: t 11 =0, if flight i 1 is the first flight in the gate; otherwise, t 11 =c ii + d ii, where ii is the flight preceding flight i 1 ; t 12 =, if flight j 1 is the last flight in the gate; otherwise, t 12 = AttemptShiftRight(k 1,jj), where jj is the flight follows flight j 1 ; t 13 = c i ; t 14 = c j + d j ; t 15 = AttemptShiftIntervalRight(k 1,i 1,j 1 ); t 16 = b j ; The values of t 21,t 22,t 23,t 24,t 25,t 26 which corresponds to the interval [i 2,j 2 ] can be calculated in the similar manner. First, we can use the values of these time points to determine some incompatible situations. If one of the following inequalities holds, we can say the two intervals are incompatible: t 21 >t 15,t 11 >t 25,t 12 < t 24,t 22 <t 14. This is because overlapping cannot be avoided even if the flights are shifted as far left (or right) as possible. We then attempt to shift the intervals [i 1,j 1 ] and [i 2,j 2 ] to the appropriate places:

16 20 A. LIM ET AL. result1 AttemptShiftInterval(k 1,i 1,j 1,t 21 ) shift the interval [i 1,j 1 ] to the place starting at time t 21 ; result1 returns its end point; result2 AttemptShiftInterval(k 2,i 2,j 2,t 11 ) shift the interval [i 2,j 2 ] to the place starting at time t 11 ; result2 returns its end point; Following this, in order to verify whether the exchange is feasible, we check if the inequalities result1 t 22 and result2 t 12 hold. That is, whether the endpoints of two intervals overlap with the original flights. Algorithm 2 The Interval Exchange Move Algorithm succ false while not succ do generate i 1,j 1,i 2,j 2 randomly, obtain the values of k 1 and k 2 calculate the values of t 11,t 12,t 13,t 14,t 15,t 16 calculate the values of t 21,t 22,t 23,t 24,t 25,t 26 if t 21 >t 15 or t 11 >t 25 or t 12 <t 24 or t 22 <t 14 then continue; end if result1 AttemptShiftInterval(k 1,i 1,j 1,t 21 ) result2 AttemptShiftInterval(k 2,i 2,j 2,t 11 ) if result1 t 22 and result2 t 12 then succ true ShiftInterval(k 1,i 1,j 1,t 21 ) ShiftInterval(k 2,i 2,j 2,t 11 ) perform the exchange move update the cost break end if end while The interval exchange move algorithm is described in Algorithm A TS algorithm TS is a meta-heuristic search procedure that proceeds iteratively from one solution to another by moves in a neighborhood space with the assistance of adaptive memory. We found the TS approach to be suited for this problem. TS memory plays an important role in the search process. It forbids solution attribute changes recorded in the short-term memory to be reused. How long a restriction is in effect

17 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 21 depends on the tabu tenure parameter, which identifies the number of iterations a particular restriction remains in force (Glover and Laguna 1997). TS consists of a neighborhood search and the use of short-term memory. In the following, we first describe TS memory and then sketch the framework used for the problem TS memory In our problem, the solutions are represented by two sequences A and T, which both have length n (n is the number of flights). The sequence A represents the gate assignments and the sequence T represents the gate open time for flights. In the TS memory we implement, only the assignment information is captured so that only the move that has the identical assignment to the assignments in TS memory will be forbidden. A TS restriction is overridden by aspiration if the outcome of the move under consideration is sufficiently desirable. The aspiration criterion is satisfied if the current move is better than the best move so far in our TS implementation TS framework The TS algorithm can be described by the following steps: 1. Generate an initial solution x init randomly, set x curr x init ; 2. Generate a set of neighborhood solutions N(x curr ) of x curr by the Insert Move and Interval Exchange Move; 3. The solution x N(x curr ) with the least cost and satisfying either one of the two conditions will be selected: (1) it is not forbidden (i.e. the assignment is not identical to any assignments of recent tabu tenure moves); (2) The cost of x is better than the current best cost (aspiration criterion); 4. Set x curr x ; update the TS memory; 5. If the termination conditions are satisfied, stop; otherwise jump to step 2; When we generate the neighborhood solutions, we randomly choose Insert Move and Interval Exchange Move with equal probability. There are two termination conditions: either the best solution cannot be improved within a certain number of iterations, or the maximum number of iterations has been reached.

18 22 A. LIM ET AL A memetic algorithm Genetic algorithms (GA) (Holland 1975) have become a well-known meta-heuristic approach for difficult combinatorial optimization problems and led to the development of MA in 1989 (Moscato 1989). In this second approach to the gate assignment problem, we found it suitable to employ a local-search based MA to solve the airport gate assignment problem. We first discuss some essential components of MA, including solution representation, crossover and mutation operators, and local search, and then outline the framework of our MA Solution representation The chromosome is an important component in MA and has great influence on the algorithm s outputs. In the basic GA, a chromosome is usually encoded as a sequence and represents a solution. In this problem, however, solutions are not easy to encode as sequences, since both the gate assignment and gate occupied time needs to be determined. Thus, in our MA, a chromosome sequence only represents the gate assignment of the solution, but the gate occupied time of all the flights are determined by a greedy algorithm. For example, consider an instance with m gates and n flights. The chromosome is a sequence (s 1,s 2,...,s n ), which means that Flight 1 is assigned to gate s 1, Flight 2 is assigned to gate s 2,..., Flight n is assigned to gate s n (1 s i m, 1 i n). Given such a chromosome sequence, we use Algorithm 3 to obtain c i (1 i n). Note that the algorithm does not guarantee finding a feasible solution. In the case where a feasible solution cannot be found, we drop the sequence and use another sequence Crossover operator In our problem, chromosomes are not permutation sequences such as in the Travelling Salesman Problem. Hence, well-known crossover operators, such as Partially Mapped Crossover (Goldberg and Lingle 1985) and Cycle Crossover (Oliver et al. 1987), cannot be used. We implemented two crossover operators: One-Point Crossover and Two-Point Crossover. In the One-Point Crossover, one random crossover point is selected. The first part of the first parent is attached with the second part of the second parent to make the first offspring. The second offspring is built from the first part of the second parent and the second part of the first parent. The following is an example of One-Point Crossover operator (the crossover point is denoted by ):

19 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 23 Algorithm 3 A Greedy Algorithm to Generate Solutions Input: A chromosome sequence S with length n for each flight i in 1..n do g S i,iter 0 while not succeed and iter <m do if assign flight i to gate g is successful then assign flight i to gate g as early as possible update S i and c i return else g g mod m + 1 iter iter + 1 end if end while end for Parent1: ( ) Parent2: ( ) Offspring1: ( ) Offspring2: ( ) In the Two-Point Crossover, two random crossover points are selected for one crossover operation. The chromosomal materials are swapped between two cut points to produce offsprings. This is illustrated in the following example: Parent1: ( ) Parent2: ( ) Offspring1: ( ) Offspring2: ( ) We conducted experiments using both of the above-mentioned operators and found that Two-Point Crossover works better than One- Point Crossover. Details of experiments are provided in the section on experiments Mutation operator We chose the Swap Mutation as our mutation operator, which selects two positions at random and swaps the values at those positions. For

20 24 A. LIM ET AL. Figure 6. Terminal topology. example, the following mutation swaps the values at position 3 and position 6: ( ) ( ) Local search enhancement Local search is applied to enhance the solutions obtained after crossover and mutation are implemented. We use the TS algorithm given above. Here, the total number of iterations in the TS algorithm and the set of generated neighborhood solutions are significantly reduced, as the TS algorithm is applied repeatedly on the individuals of each generation. Taking the time into consideration, we only apply local search on a subset (20% in our experiments) of individuals of each generation MA framework With these components of MA, we now outline Algorithm 4. In this algorithm, #pop, #crossover, #iter and p 1,p 2 are parameters which are specified within experiments. 5. Experimental Results All the algorithms were coded in Java and run on a Pentium IV 1.6 GHz machine. As comparison, we use CPLEX to solve the formulation presented in Section 2. The test generation process, parameter settings of various methods, and detailed computational results are presented in the following subsections.

21 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 25 Algorithm 4 Memetic Algorithm Initialize Pop with size #pop for iter 1to#iter do for off 1to#crossover do Randomly select ParentA and ParentB Crossover ParentA and ParentB to produce OffspringA and OffspringB end for for each new-produced individual indv do mutate indv with probability p 1 call tabusearch(indv) with probability p 2 end for evaluate each individual select the best #pop individuals from all the individuals update current best solution end for 5.1. Test data generation We chose a representative layout of an airport to have two parallel sets of terminals, where gates are symmetrically located in the two terminals shown in Figure 6. Such a scheme is not uncommon in airports today. We set the distance between two adjacent gates within one terminal (e.g., gate 1 and gate 3) to be 1 unit and the distance between two parallel gates in different terminals (e.g., gate 1 and gate 2) to be 3 units. To simplify the problem, we assumed that passengers can only walk horizontally or vertically, i.e., if one passenger wants to transfer from gate 3 to gate 2, his walking distance is 1+3=4 units (thus the distances are rectilinear). This is similar to the so-called Manhattan metric. We note that it is easy to use other terminal topologies including other metrics for the problem to compare performance. The best topologies would be those that give the smallest objective values. The test data generation program requires two parameters: The number of flights n and the number of gates m. The n start points of flight time window a i (1 i n) were uniformly generated in the interval [1, n 70 m ]. The end points of flight i time window b i were generated as b i =a i +[45, 74]. The gate occupied duration d i is calculated as d i = (b i a i ) des, where des is percentage parameter controlling the level of gate occupied durations against time windows. For example, when des= 100%, the problem becomes the classical airport gate assignment problem. The smaller the des value is, the more flexible it is to adjust the gate occupied time. The unit delay penalty p i was generated in the

22 26 A. LIM ET AL. interval [10, 14]. The rectilinear distance between gates w kl is calculated in the above-mentioned method and the number of transferring passengers f ij is randomly generated in the interval [6, 60] if a i <a j (0 otherwise) Experimental setup and parameter settings We conducted a series of experiments to test the performance of our heuristic approaches. The following four types of algorithms are implemented: Tabu Search (TS): Maximum number of iterations is 10 6 and each time 100 neighbors are generated with a tabu tenure = 10. The algorithm is to terminate if the best solution was not improved within 10 4 iterations. Memetic Algorithm (MA): The following values are used: #iter = 10 4, #pop = 300, #crossover = 500. The mutation probability p 1 is taken to be 0.2 and the local search probability p 2 is taken to be 0.2. The algorithm terminated if the best solution is not improved within 50 iterations. Two variations are actually compared here: MA1 uses the One-Point Crossover operator and MA2 uses the Two-Point Crossover operator. Genetic Algorithm (GA): The algorithm is similar to MA except that there is no local search component involved. The parameters settings are the same as in MA, but the maximum iteration is 10 5 and the termination condition was when the best solution did not improve within 500 iterations. The reason we increase the values of these two parameters is because GA runs faster than MA and we attempt to compare them in the similar running time. We also implement GA1 and GA2 which use the different crossover operators The results We designed three categories of instances to test our algorithms. The detailed results are presented in Table 1 3, respectively. Categories 1 and 2 consist of 10 instances. Category 3 has 40 instances in 8 groups. The first row of each table denotes the instance ID or group ID. The second row contains the sizes, where n m means that there are n aircrafts and m gates. The rest of the rows provide the results of various methods proposed in this paper. Each result cell contains two values.

23 AIRPORT GATE SCHEDULING WITH TIME WINDOWS 27 The value on the top provides the result, whereas the value at the bottom provides the computational time in seconds. 1. Small Sizes Instances Ten small scale instances are generated with the sizes (n m) ranging from 12 3to15 4. Parameter des was set to be 0.7. We used these instances to test the performance of CPLEX solver and our heuristic algorithms. The results are shown in Table 1. We found that all the heuristic algorithms can obtain optimal solutions with the running times much less than those needed by CPLEX solver in most of the time. 2. Medium Sizes Instances Ten instances which range from 20 5to28 6 are used here. Parameter des was set to be 0.7. We set the running time limit of CPLEX solver to be 1800 s. With these settings, the solutions obtained by CPLEX are not guaranteed to be optimal solutions. The results are presented in Table 2. It is clear that the heuristic algorithms outperform CPLEX solver greatly and with much shorter running time. CPLEX solver could not obtain any feasible solutions for instance 9. Among these heuristic algorithms, the results are close. TS performs slightly better than others with the similar running time. 3. Large Sizes Instances Forty large scale instances are generated and categorized into 8 groups with the size ranging from to and each group contains 5 instances. Parameter des is randomly generated from 0.6 to 0.8. The results can be found in Table 3. TS is no Table 1. Results of CPLEX and heuristics on random instances with small sizes Instance: Size: CPLEX Time TS Time MA Time MA Time GA Time GA Time

24 28 A. LIM ET AL. Table 2. Results of CPLEX and heuristics on random instances with medium sizes Instance: Size: CPLEX Time TS Time MA Time MA Time GA Time GA Time Table 3. Results of heuristics on random instances with large sizes Group: Size: TS Time MA Time MA Time GA Time GA Time doubt the best performer here. It outperforms all the other algorithms significantly and with much less running time. Comparing with MA and GA, we observe that MA performs better than GA with similar or smaller running times. Also, there are no significant difference on the results between the two crossover operators. Our heuristics outperformed the IP-based model using the CPLEX solver in terms of quality of solutions and computing times needed. The superiority of our methods is more apparent in larger test instances. From the observations from the experiments, we believe that our neighborhood search is effective and TS is a suitable approach to tackle the proposed airport gate assignment problem with time windows. MA, though not as good as TS, is a good approach to this problem with the aid of local search component.

25 AIRPORT GATE SCHEDULING WITH TIME WINDOWS Applications to Cross-Docking As we have seen in Section 3, the model studied here can be extended to address handling of material other than passengers. There has been a number of studies on cross-docking recently. Tsui and Chang (1992) used a bilinear program of assigning trailers to doors, where the objective was to minimize weighted distances between incoming and outgoing trailers. Recently, a study by Bartholdi and Gue (2000) examined minimizing labor costs in freight terminals by properly assigning incoming and outgoing trailers to doors. The authors highlight some similarities between gate assignments in airports and layout in freight terminals. As in the case of truck arrivals at freight terminals, aircraft arrival times can vary. As we have seen, like freight, when considered altogether, passengers, baggage and cargo can be very heterogeneous, each requiring specialized material handling. Also, as in freight terminals, airports can become congested in periods of time within localized areas. Although cross-docking studies such as that by Bartholdi and Gue (2000) have considered intra-terminal factors such as types of congestion that impact costs, they do not address actual gate assignments to arriving vehicles when arrival times can change. The model developed and solved here can be applied to such situations to improve or augment cross-docking optimization. 7. Conclusion In this work, we studied the airport gate assignment problem with time windows. In contrast to the existing airport gate assignment studies, where flight have fixed schedules, we consider the more realistic situation where flight arrival and departure times can change. Although we minimize walking distances (or travel time) in our objective function, the model can be easily adapted for optimizing other material handling costs including baggage and cargo costs. This is achieved through gate assignments where time slots alloted to aircraft at gates are closest possible to scheduled time slots. Further, the model can be applied to cross-docking optimization in areas other than airports such as freight terminals, where material arrival times (via trucks, ships) can fluctuate. This work augments work done on cross-docking optimization. The solution approach uses insert and interval exchange moves together with a time shift algorithm. We then

26 30 A. LIM ET AL. use these neighborhood moves in TS and MA. Computational results are provided, and verify that our heuristics work well in small cases and much better in large cases when compared with CPLEX solver. Acknowledgements The authors thank the anonymous referees for invaluable suggestions, which helped improve the content of this work and its presentation. The third-named author acknowledges support for this work from the Wharton-SMU Research Center, Singapore Management University. References Babic, O., Teodorovic, D. & Tosic, V. (1984). Aircraft Stand Assignment to Minimize Walking. Journal of Transportation Engineering 110: Baron, P. (1969). A Simulation Analysis of Airport Terminal Operations. Transportation Research 3: Bartholdi, J., & Gue, K. (2000). Reducing Labor Costs in an LTL Cross-docking Terminal. Operations Research 48: Bihr, R. (1990). A Conceptual Solution to the Aircraft Gate Assignment Problem Using 0,1 Linear Programming. Computers & Industrial Engineering 19: Braaksma, J. & Shortreed, J. (1971). Improving Airport Gate Usage with Critical Path Method. Transportation Engineering Journal of ASCE Cheng, Y. (1998a). Network-based Simulation of Aircraft at Gates in Airport Terminals. Journal of Transportation Engineering Cheng, Y. (1998b). A Rule-based Reactive Model for the Simulation of Aircraft on Airport Gates. Knowledge-based Systems 10: Ding, H., Lim, A., Rodrigues, B. & Zhu, Y. (2003a). New Heuristics for the Over-constrained Flight to Gate Assignments. Journal of the Operational Research Society, to appear. Ding, H., Lim, A., Rodrigues, B. & Zhu, Y. (2003b). The Over-constrained Airport Gate Assignment Problem. Computers & Operational Research, to appear. Glover, F. & Laguna, M. (1997). Tabu Search. Kluwer Acadamic Publishers. Goldberg, D. & Lingle, R. (1985). Alleles, Loci, and The Traveling Salesman Problem. Haghani, A. & Chen, M.-C. (1998). Optimizing Gate Assignments at Airport Terminals. Transportation Research A 32(6): Holland, J. (1975). Adaptation in Natural and Artifical Systems. The University of Michigan Press. Mangoubi, R. & Mathaisel, D. (1985). Optimizing Gate Assignments at Airport Terminals. Transportation Science 19: Moscato, P. (1989). On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms. Technical Report, Caltech Concurrent Computation Program, C#P Report 826.